It cannot be proven because it is not true. You can prove that by choosing x and y that satisfies sin2x=3sin2y. For example, take x = 15 degrees and y = 9.5941 degrees.
For those choices, sin 2x = 3sin 2y = 0.500
For those choices of x and y
2 tan(x-y) = 0.1893 and tan (x+y) = 0.4577

By being able to find ANY x,y combination that satisfies sin2x=3sin2y, and for which your second equation is not true, then your contention is disproved.

Either someone is playing a practical joke on us, or on you, or you have copied the problem incorrectly